Different animations for different scenes for a linked armature
When you link an Object from one Scene to another all changes are transferred from one Scene to another, even animations. That is because the same Object and Armature are used. There is a quick way of fixing this for your case: Add a new armature object in each scene, then select each one of them and make sure they use the same Armature. You can make the selection here:


All changes to the Armature in Edit Mode will be transferred to all other scenes but changes made in the Pose Mode will be Scene specific.